Dynamic monopolies for degree proportional thresholds in connected graphs of girth at least five and trees

For a connected graph G of maximum degree at least 1ρ, Chang showed hρ(G)≤5.83ρn(G), which was improved by Chang and Lyuu to hρ(G)≤4.92ρn(G). We show that for every ϵ>0, there is some ρ(ϵ)∈(0,1) such that hρ(G)≤(2+ϵ)ρn(G) for every ρ in (0,ρ(ϵ)), and every connected graph G that has maximum degree at least 1ρ and girth at least 5. Furthermore, we show that hρ(T)≤ρn(T) for every ρ in (0,1], and every tree T that has order at least 1ρ.